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AMC8 2025

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AMC8 · 2025

Q1
The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire $4\times4$ grid is covered by the star?
下图所示的八角星是一种流行的绗缝图案。八角星占整个 $4\times4$ 网格的百分之多少？
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Q2
The table below shows the ancient Egyptian hieroglyphs that were used to represent different numbers. For example, the number $32$ was represented by the hieroglyphs $\cap \cap \cap ||$. What number is represented by the following combination of hieroglyphs?
下表展示了古埃及象形文字用于表示不同数字的符号。 例如，数字 $32$ 用象形文字 $\cap \cap \cap ||$ 表示。以下象形文字组合表示的数字是多少？
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Q3
Buffalo Shuffle-o is a card game in which all the cards are distributed evenly among all players at the start of the game. When Annika and 3 of her friends play Buffalo Shuffle-o, each player is dealt 15 cards. Suppose 2 more friends join the next game. How many cards will be dealt to each player?
Buffalo Shuffle-o 是一种纸牌游戏，游戏开始时所有牌平均分发给所有玩家。当 Annika 和她的 3 个朋友玩时，每位玩家分到 15 张牌。假如再有 2 个朋友加入下一局。每位玩家将分到多少张牌？
Q4
Lucius is counting backward by $7$s. His first three numbers are $100$, $93$, and $86$. What is his $10$th number?
Lucius 按 $7$ 倒数计数。他的前三个数是 $100$、$93$ 和 $86$。他的第 $10$ 个数是多少？
Q5
Betty drives a truck to deliver packages in a neighborhood whose street map is shown below. Betty starts at the factory (labled $F$) and drives to location $A$, then $B$, then $C$, before returning to $F$. What is the shortest distance, in blocks, she can drive to complete the route?
Betty 开卡车在社区送包裹，下图是街道图。Betty 从工厂（标为 $F$）出发，依次开往位置 $A$、$B$、$C$，然后返回 $F$。她完成路线的最短距离（以街区为单位）是多少？
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Q6
Sekou writes the numbers $15, 16, 17, 18, 19.$ After he erases one of his numbers, the sum of the remaining four numbers is a multiple of $4.$ Which number did he erase?
Sekou 写下数字 $15, 16, 17, 18, 19$。他擦掉其中一个数字后，剩下四个数字的和是 $4$ 的倍数。他擦掉了哪个数字？
Q7
On the most recent exam on Prof. Xochi's class, - 5 students earned a score of at least $95\%$, - 13 students earned a score of at least $90\%$, - 27 students earned a score of at least $85\%$, - 50 students earned a score of at least $80\%$, How many students earned a score of at least $80\%$ and less than $90\%$?
在 Xochi 教授班级最近一次考试中， - 5 名学生的成绩至少为 $95\%$， - 13 名学生的成绩至少为 $90\%$， - 27 名学生的成绩至少为 $85\%$， - 50 名学生的成绩至少为 $80\%$， 有多少名学生的成绩至少为 $80\%$ 且低于 $90\%$？
Q8
Isaiah cuts open a cardboard cube along some of its edges to form the flat shape shown on the right, which has an area of $18$ square centimeters. What is the volume of the cube in cubic centimeters?
Isaiah 将一个纸板立方体沿一些边切割开来，形成右侧所示的平面形状，其面积为 $18$ 平方厘米。这个立方体的体积是多少立方厘米？
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Q9
Ningli looks at the $6$ pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting $6$ numbers?
Ningli 查看时钟上直接相对的 $6$ 对数字。她计算每对数字的平均值。所得 $6$ 个数的平均值是多少？
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Q10
In the figure below, $ABCD$ is a rectangle with sides of length $AB = 5$ inches and $AD = 3$ inches. Rectangle $ABCD$ is rotated $90^\circ$ clockwise around the midpoint of side $DC$ to give a second rectangle. What is the total area, in square inches, covered by the two overlapping rectangles?
下图中，$ABCD$ 是长 $AB = 5$ 英寸、高 $AD = 3$ 英寸的矩形。矩形 $ABCD$ 绕边 $DC$ 中点顺时针旋转 $90^\circ$ 得到第二个矩形。两个重叠矩形覆盖的总面积是多少平方英寸？
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Q11
A $\textit{tetromino}$ consists of four squares connected along their edges. There are five possible tetromino shapes, $I$, $O$, $L$, $T$, and $S$, shown below, which can be rotated or flipped over. Three tetrominoes are used to completely cover a $3\times4$ rectangle. At least one of the tiles is an $S$ tile. What are the other two tiles?
一种\textit{四格子}由四个正方形沿边连接而成。有五种可能的四格子形状，$I$、$O$、$L$、$T$和$S$，如下所示，可以旋转或翻转。使用三个四格子完全覆盖一个$3\times4$矩形。至少有一个是$S$格子。另外两个格子是什么？
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Q12
The region shown below consists of 24 squares, each with side length 1 centimeter. What is the area, in square centimeters, of the largest circle that can fit inside the region, possibly touching the boundaries?
下图所示区域由24个边长1厘米的正方形组成。能放入该区域内最大圆的面积是多少平方厘米，该圆可能触及边界？
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Q13
Each of the even numbers $2, 4, 6, \ldots, 50$ is divided by $7$. The remainders are recorded. Which histogram displays the number of times each remainder occurs?
每个偶数$2, 4, 6, \ldots, 50$除以$7$。记录余数。哪个直方图显示了每个余数出现的次数？
Q14
A number $N$ is inserted into the list $2$, $6$, $7$, $7$, $28$. The mean is now twice as great as the median. What is $N$?
一个数$N$插入列表$2$、$6$、$7$、$7$、$28$中。现在平均数是中位数的两倍。$N$是多少？
Q15
Kei draws a $6$-by-$6$ grid. He colors $13$ of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let $m$ and $M$ equal the least and greatest possible number of gold-on-gold pairs, respectively. What is the value of $m+M$?
Kei画了一个$6$×$6$网格。他将$13$个单元正方形涂成银色，其余涂成金色。然后他沿垂直方向对折网格，形成重叠的单元正方形对。让$m$和$M$分别是金对金对的最小和最大可能数。$m+M$的值是多少？
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Q16
Five distinct integers from $1$ to $10$ are chosen, and five distinct integers from $11$ to $20$ are chosen. No two numbers differ by exactly $10$. What is the sum of the ten chosen numbers?
从$1$到$10$中选择5个不同的整数，从$11$到$20$中选择5个不同的整数。没有任何两个数相差恰好$10$。所选十个数的和是多少？
Q17
In the land of Markovia, there are three cities: $A$, $B$, and $C$. There are 100 people who live in $A$, 120 who live in $B$, and 160 who live in $C$. Everyone works in one of the three cities, and a person may work in the same city where they live. In the figure below, an arrow pointing from one city to another is labeled with the fraction of people living in the first city who work in the second city. (For example, $\frac{1}{4}$ of the people who live in $A$ work in $B$.) How many people work in $A$?
在马尔科维亚，有三个城市：$A$、$B$和$C$。$A$有100人居住，$B$有120人，$C$有160人。每人都在一个城市工作，可以在居住的城市工作。下图中，从一个城市指向另一个城市的箭头标有居住在第一个城市而在第二个城市工作的人的 fraction。（例如，居住在$A$的$\frac{1}{4}$的人在$B$工作。）有多少人在$A$工作？
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Q18
The circle shown below on the left has a radius of 1 unit. The region between the circle and the inscribed square is shaded. In the circle shown on the right, one quarter of the region between the circle and the inscribed square is shaded. The shaded regions in the two circles have the same area. What is the radius $R$, in units, of the circle on the right?
左边的圆半径为1单位。圆与内接正方形之间的区域被涂影。右边的圆中，圆与内接正方形之间的区域的四分之一被涂影。两个圆的涂影区域面积相等。右边圆的半径$R$是多少单位？
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Q19
Two towns, $A$ and $B$, are connected by a straight road that is $15$ miles long. Travelling from city $A$ to town $B$, the speed limit changes every $5$ miles: from $25$ to $40$ to $20$ miles per hour (mph). Two cars, one at town $A$ and one at town $B$, start moving toward each other at the same time. They drive at exactly the speed limit in each portion of the road. How far from town $A$, in miles, will the two cars meet?
两个城镇$A$和$B$由一条15英里长的直路连接。从$A$到$B$，每5英里限速变化：25、40、20英里每小时（mph）。两辆车，一辆在$A$，一辆在$B$，同时开始向对方行驶。它们在每段路严格按限速行驶。两车将在距离$A$镇多少英里处相遇？
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Q20
Sarika, Dev, and Rajiv are sharing a large block of cheese. They take turns cutting off half of what remains and eating it: first Sarika eats half of the cheese, then Dev eats half of the remaining half, then Rajiv eats half of what remains, then back to Sarika, and so on. They stop when the cheese is too small to see. About what fraction of the original block of cheese does Sarika eat in total?
Sarika、Dev和Rajiv分享一大块奶酪。他们轮流切下剩余奶酪的一半并吃掉：先Sarika吃一半，然后Dev吃剩余一半的一半，然后Rajiv吃剩余的一半，然后回到Sarika，依此类推。当奶酪太小时停止。Sarika总共吃了原奶酪的大约几分之几？
Q21
The Konigsberg School has assigned grades 1 through 7 to pods $A$ through $G$, one grade per pod. Some of the pods are connected by walkways, as shown in the figure below. The school noticed that each pair of connected pods has been assigned grades differing by 1 or more grade levels. (For example, grades 1 and 2 will not be in pods directly connected by a walkway.) What is the sum of the grade levels assigned to pods $C, E$, and $F$?
柯尼斯堡学校将1到7的年级分配给荚A到G，每个荚一个年级。一些荚通过人行道连接，如下图所示。学校注意到，每对相连的荚分配的年级相差1个或更多年级。（例如，1和2年级不会在直接连接的人行道荚中。）C、E和F荚分配的年级水平之和是多少？
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Q22
A classroom has a row of 35 coat hooks. Paulina likes coats to be equally spaced, so that there is the same number of empty hooks before the first coat, after the last coat, and between every coat and the next one. Suppose there is at least 1 coat and at least 1 empty hook. How many different numbers of coats can satisfy Paulina's pattern?
一个教室有一排35个衣帽钩。Paulina喜欢外套等间距放置，使得第一个外套前、最后一个外套后以及每个外套与下一个外套之间都有相同数量的空钩。假设至少有1件外套且至少有1个空钩。有多少种不同的外套数量可以满足Paulina的模式？
Q23
How many four-digit numbers have all three of the following properties? (I) The tens and ones digit are both 9. (II) The number is 1 less than a perfect square. (III) The number is the product of exactly two prime numbers.
有多少个四位数具有以下三个性质？ (I) 十位和个位都是9。 (II) 该数比一个完全平方少1。 (III) 该数恰好是两个素数的乘积。
Q24
In trapezoid $ABCD$, angles $B$ and $C$ measure $60^\circ$ and $AB = DC$. The side lengths are all positive integers, and the perimeter of $ABCD$ is 30 units. How many non-congruent trapezoids satisfy all of these conditions?
在梯形$ABCD$中，角$B$和$C$分别测得$60^\circ$且$AB = DC$。边长均为正整数，且$ABCD$的周长为30单位。满足所有这些条件的非全等的梯形有多少个？
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Q25
Makayla finds all the possible ways to draw a path in a $5 \times 5$ diamond-shaped grid. Each path starts at the bottom of the grid and ends at the top, always moving one unit northeast or northwest. She computes the area of the region between each path and the right side of the grid. Two examples are shown in the figures below. What is the sum of the areas determined by all possible paths?
Makayla找出在$5 \times 5$菱形网格中绘制路径的所有可能方法。每条路径从网格底部开始，到顶部结束，总是向东北或西北移动一个单位。她计算每条路径与网格右侧之间的区域面积。下图显示了两个例子。所有可能路径确定的面积之和是多少？
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